Lecture 15 Estimating a Population Proportion

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Lecture 15 Estimating a Population Proportion Powered By Docstoc
					         Sections 7.1 and 7.2

 This chapter presents the beginning
 of inferential statistics.
 The two major applications of inferential
statistics

  Estimate a population parameter: proportion, mean
  Test some claim (or hypothesis) about a population.
Point estimate: a single number
Interval estimate: interval of numbers.
       Confidence Interval

 Why?: point estimate is not reliable under
re-sampling.
 A confidence interval (CI): an interval of
values used to estimate the true population
parameter.
              Point Estimate

   p=          population proportion

   ˆ= n
   p  x         sample proportion
                of x successes in a sample of size n.
(pronounced     Unbiased estimate (best estimate)
   ‘p-hat’)

  ˆ       ˆ
  q = 1 - p = sample proportion
                 of failures in a sample size of n
Example: Photo-Cop Survey Responses
829 adult Minnesotans were surveyed, and 51% of them are
opposed to the use of the photo-cop for issuing traffic
tickets. Using these survey results, find the best estimate of
the proportion of all adult Minnesotans opposed to photo-
cop use.

Best point estimate=sample proportion=51%.
         Confidence Level

 α: between 0 and 1
A confidence level: 1 - α or 100(1- α)%. E.g. 95%.
This is the proportion of times that the confidence
interval actually does contain the population parameter,
assuming that the estimation process is repeated a
large number of times.
Other names: degree of confidence or the
confidence coefficient.
The Critical Value
    (z-score)
                     Given α
Finding zα/2 for 100(1- α)% Confidence Level

                                 α =5%


                               α/2 = 2.5% = .025
          Sampling Distribution of ^
                                   p
The sampling distribution of sample proportion can be
approximated by a normal distribution if np≥15 and
 nq ≥15 : phat is approximately N(p, pq/n), q=1-p.
                        p             p− p
                                      ˆ
                                   z=
                                        ˆˆ
                                        pq
                                        n




                        p
                                        ^
                                        p
                    ^
 Margin of Error of p
the maximum likely (with probability 1 – α)
difference between the observed proportion
^
p and the true population proportion p.


       E = zα / 2
                            pˆ
                            ˆq
                             n

                                        ^
                       Standard Error of p
                       =se
Finding the 95% Confidence Interval
    for a Population Proportion
 A 95% confidence interval for a population
 proportion p is:

                                     ˆ     ˆ
                                     p(1 - p)
             p ± 1.96(se), with se =
             ˆ
                                        n
 100(1-α)% confidence interval for p is

                                             p(1 − p)
                                             ˆ     ˆ
            p ± zα / 2 ( se)
            ˆ                  with   se =
                                                n
Example: Would You Pay Higher
Prices to Protect the Environment?
In 2000, the GSS asked: “Are you willing to
pay much higher prices in order to protect
the environment?”
  Of n = 1154 respondents, 518 were willing to
  do so
Find and interpret a 95% confidence interval
for the population proportion of adult
Americans willing to do so at the time of the
survey
Example: Would You Pay Higher
Prices to Protect the Environment?


         518
    p=
    ˆ         = 0.45
        1154
           (0.45)(0.55)
    se =                = 0.015
               1154
     E = 1.96(se) = 1.96(0.015) = 0.03
    p ± E = 0.45 ± 0.03 = (0.42, 0.48)
    ˆ
What is the Error Probability for the
  Confidence Interval Method?
Summary: Effects of Confidence Level
 and Sample Size on Margin of Error

 The margin of error for a confidence interval:

   Increases as the confidence level increases
   Decreases as the sample size increases
           Determining Sample Size
Recall :
           E=    zα / 2       ˆˆ
                              pq
                              n

                    (solve for n by algebra)



            n=    zα / 2 p q
                         ˆˆ
                          2

                     E2
          Sample Size for Estimating
                Proportion p

                    ˆ
  When an estimate p of p is known:


      n=    ( zα / 2   )2
                            ˆˆ
                            pq
                   E2
When no estimate of p is known:

           ( zα / 2)2 0.25
     n=           E2
Example: Suppose a sociologist wants to determine       the
current percentage of U.S. households using e-mail. How many
households must be surveyed in order to be 95% confident that the
sample percentage is in error by no more than four percentage
points?

a) Use this result from an earlier study: In 1997, 16.9% of U.S.
   households used e-mail (based on data from The World Almanac
   and Book of Facts).
b) Assume that we have no prior information suggesting a possible
   value of p.
a) Use this result from an earlier study: In 1997, 16.9% of U.S.
   households used e-mail (based on data from The World Almanac
   and Book of Facts).




n = [za/2 ]2ˆ ˆq
             p                      To be 95% confident that our
      E2                            sample percentage is within
                                    four percentage points of the
 = [1.96]2 (0.169)(0.831)           true percentage for all
         0.042                      households, we should
                                    randomly select and survey
 = 337.194                          338 households.
 = 338 households
 b) Assume that we have no prior information suggesting a possible
    value of p.



n = [za/2 ]2 • 0.25
      E2                      With no prior information,
 = (1.96)2 (0.25)             we need a larger sample to
      0.042                   achieve the same results
                              with 95% confidence and an
 = 600.25                     error of no more than 4%.
 = 601 households
          Finding the Point Estimate
                and E from a
             Confidence Interval
Point estimate of p:
ˆ
p = (upper confidence limit) + (lower confidence limit)
                                  2

Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
                                  2

				
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